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This paper is divided into three parts. The first part focuses on periodiclayer heat potentials, demonstrating their smooth dependence on regularperturbations of the support of integration. In the second part, we present anapplication of the results from the first part. Specifically, we consider atransmission problem for the heat equation in a periodic two-phase compositematerial and we show that the solution depends smoothly on the shape of thetransmission interface, boundary data, and conductivity parameters. Finally, inthe last part of the paper, we fix all parameters except for the contrastparameter and outline a strategy to deduce an explicit expansion of thesolution using a Neumann-type series.

Multi-parameter perturbations for the space-periodic heat equation